Have You Tried Rearranging The Equation?

Logic Level 1

□ ÷ □ = □ ÷ □ \large \square \; \div \; \square \; = \; \square \; \div \; \square

If all but one of the numbers {2, 3, 4, 5, 6} were used in the equation above, which one wasn't used?

2 3 4 5 6

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Pi Han Goh by Pi Han Goh , 30, Malaysia

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3 solutions

Jolon Behrent
Aug 13, 2016

Relevant wiki: Arithmetic Puzzles - Fill in the Blanks

6 3 = 4 2 \frac{6}{3}=\frac{4}{2}

We can prove this by seeing that out of the 5 numbers we can choose, we have 3 primes: 2, 3, and 5. 2 can be multiplied into 4 and 6, 3 can go into 6, and 5 can't go into any number (next multiple of 5 is 10).

As 3 is a factor of only 6, and 2 is a factor of both 4 and 6, we can see that the answer must be 6 3 = 4 2 \frac{6}{3}=\frac{4}{2}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Another way to explain it is to show that, by cross multiplication, the given equation is equivalent to

□ × □ = □ × □ . \large \square \; \times \; \square \; = \; \square \; \times \; \square .

If we place "5" inside one of these cells, then we cannot use "5" any more because the four numbers in the four squares are distinct by assumption. So only one side of the equation is divisible by 5, while the other side is not, which implies that "5" cannot be used in the given equation. What's left to do is to show that all the remaining numbers can be used to make the equation true.

THANKS OBAMA

Pi Han Goh - 4 years, 10 months ago

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I used it the other way around: 2/3 = 4/6

Peter van der Linden - 4 years, 10 months ago
Razzi Masroor
Aug 18, 2016

For bigger sets when you ask this question, the numbers that wont be used and primes that are greater than half of the biggest number. First if you had a prime like that in a set, it couldn't be a numerator.It also cant be a denominator since its prime and it would have no number that is 2x greater, thats the only way a number could work,there would be 2 numerators and denominators where one numerator and denominator is atleast a positive integer except for 1x more than the other set. So 5 cant work even though six does since it is 2x3

1 Helpful 0 Interesting 0 Brilliant 0 Confused

{6/3=4/2). Number that wasn't used is 5.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

If it could be used then there is an answer missing... logic or not?

Peter van der Linden - 4 years, 10 months ago

Yup. Can you show that "5" can never be used?

Pi Han Goh - 4 years, 10 months ago

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